Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $z \neq 0$. $q = \dfrac{z^2 - 5z}{z^2 + 11z + 18} \times \dfrac{5z + 45}{z - 5} $
Answer: First factor the quadratic. $q = \dfrac{z^2 - 5z}{(z + 9)(z + 2)} \times \dfrac{5z + 45}{z - 5} $ Then factor out any other terms. $q = \dfrac{z(z - 5)}{(z + 9)(z + 2)} \times \dfrac{5(z + 9)}{z - 5} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac{ z(z - 5) \times 5(z + 9) } { (z + 9)(z + 2) \times (z - 5) } $ $q = \dfrac{ 5z(z - 5)(z + 9)}{ (z + 9)(z + 2)(z - 5)} $ Notice that $(z - 5)$ and $(z + 9)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac{ 5z(z - 5)\cancel{(z + 9)}}{ \cancel{(z + 9)}(z + 2)(z - 5)} $ We are dividing by $z + 9$ , so $z + 9 \neq 0$ Therefore, $z \neq -9$ $q = \dfrac{ 5z\cancel{(z - 5)}\cancel{(z + 9)}}{ \cancel{(z + 9)}(z + 2)\cancel{(z - 5)}} $ We are dividing by $z - 5$ , so $z - 5 \neq 0$ Therefore, $z \neq 5$ $q = \dfrac{5z}{z + 2} ; \space z \neq -9 ; \space z \neq 5 $